Yuichi Harikane, Ryan L. Sanders, Richard Ellis, Tucker Jones, Masami Ouchi, Nicolas Laporte, Guido Roberts-Borsani, Harley Katz, Kimihiko Nakajima, Yoshiaki Ono, Mansi Gupta

We present a JWST and ALMA detailed study of the ISM properties of high-redshift galaxies. Our JWST/NIRSpec IFU spectroscopy targeting three galaxies at $z=6-7$ detects key rest-frame optical emission lines, allowing us to derive [OII]$\lambda\lambda$3726,3729-based electron densities of $n_\mathrm{e,optical}\sim1000$ cm$^{-3}$ on average and [OIII]$\lambda$4363-based metallicities of $\mathrm{12+log(O/H)}=8.0-8.2$ in two galaxies. New ALMA Band 9/10 observations detect the [OIII]52$\mu$m line in one galaxy but do not in the others, resulting in FIR-based densities of $n_\mathrm{e,FIR}\lesssim500$ cm$^{-3}$ from the [OIII]52$\mu$m/[OIII]88$\mu$m ratios, systematically lower than the optical [OII]-based measurements. These low FIR-based densities are comparable to those at both $z\sim0$ and $z>6$ in the literature, including JADES-GS-z14-0 at $z=14.18$, suggesting little evolution up to $z\sim14$, in contrast to the increasing trend of optical-based densities with redshift. By conducting a JWST and ALMA joint analysis using emission lines detected with both telescopes, we find that the observed FIR [OIII]52,88$\mu$m luminosities are too high to be explained by the optical-based densities at which they would be significantly collisionally de-excited. Instead, a 2-zone model with distinct high- and low-density regions is required to reproduce all observed lines, indicating that FIR [OIII] emission arises predominantly from low-density gas, while optical [OIII] and [OII] lines trace both regions. We further demonstrate that the direct-$T_\mathrm{e}$ method can sometimes significantly underestimate metallicities up to 0.8 dex due to the presence of the low-density gas not fully traced by optical lines alone, highlighting the importance of combining optical and FIR lines to accurately determine gas-phase metallicities in the early universe.